In this chapter we show that there do indeed exist 3-transposition groups of type M(22), M(23), and M(24). In addition we prove the uniqueness of M(22), M(23), and M(24)′ as groups with a suitable involution centralizes Indeed we use the uniqueness results to establish the existence of the Fischer groups.

More precisely in 32.4 in [SG] it is shown that there exists a subgroup X of the Monster such that E(X) is quasisimple with center Z of order 3, X is the split extension of E(X) by an involution inverting Z, and E(X)/Z and X/Z are groups of type F24 and Aut(F24), respectively, in the sense of Sections 34 and 35 or the Introduction to Part II. In Theorem 35.1 we show each group of type Aut(F24) is generated by 3-transposition and is of type M(24). Hence Fischer's three 3-transposition groups exist, and, by Fischer's Theorem, groups of type Aut(F24) are unique up to isomorphism.

We characterize the Fischer groups (and M(24)′; = F24) via the centralizer of an involution by showing each group of type is of type M, for M = M(22) and M(23), and by showing all groups of type F24 are isomorphic. See the Introduction to Part II for an outline of the proofs of these results.